A143449 Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=6.

1, 3, 5, 7, 9, 11, 13, 15, 21, 31, 45, 63, 85, 111, 141, 183, 245, 335, 461, 631, 853, 1135, 1501, 1991, 2661, 3583, 4845, 6551, 8821, 11823, 15805, 21127, 28293, 37983, 51085, 68727, 92373, 123983, 166237, 222823, 298789, 400959, 538413, 723159, 971125

Offset: 0

Author

Alois P. Heinz, Aug 16 2008

Keywords

• nonn
• easy

Comments

a(n) is also the number of length n ternary words with at least 6 0-digits between any other digits.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=13, 3*a(n-13) equals the number of 3-colored compositions of n with all parts >=7, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,2).

Crossrefs

6th column of A143453.

Programs

Maple

a:= proc(k::nonnegint) local n,i,j; if k=0 then unapply(3^n,n) else unapply((Matrix(k+1, (i,j)-> if (i=j-1) or j=1 and i=1 then 1 elif j=1 and i=k+1 then 2 else 0 fi)^(n+k))[1,1], n) fi end(6): seq(a(n), n=0..58);

Mathematica

Series[1/(1-x-2*x^7), {x, 0, 58}] // CoefficientList[#, x]& // Drop[#, 6]& (* Jean-François Alcover, Feb 13 2014 *)

Formula

G.f.: 1/(x^6*(1-x-2*x^7)).

a(n) = 2n+1 if n<=7, else a(n) = a(n-1) + 2a(n-7). - Milan Janjic, Mar 09 2015